The dynamic figure below will help you find how to determine the maximum of ∠*BEC*. It shows a circle constructed with its centre *I* on the perpendicular bisector of *BC*, and tangent to the line segment *AD* at *G*. Drag *I* till the circle passes through *B* and *C*.

Now drag *E* to coincide with *G* while carefully watching what happens to ∠*BEC*. What do you notice? Change the shape of the trapezium *ABCD*, repeat the preceding process, and check your observation again.

1) Can you explain (prove) why the maximum of the angle has to be located at the position of *E* you identified?

2) Can you figure out a way via construction, not using experimentation and dragging as with this sketch, to precisely locate where the centre *I* of the circle should be located to determine a maximum for ∠*BEC*?

Maximizing the Angle

*Hints*:

1) Try using the exterior angle theorem to prove why the maximum of angle *BEC* is located where it is.

2) Consider the locus of points equidistant from vertex *B* and line *AD* and/or the locus of points equidistant from vertex *C* and line *AD*.

Check your explorations above, by reading my 2015 paper in *At Right Angles* and *Learning and Teaching Mathematics* at *"Flashback to the Past: a 1949 Matric Geometry Question"*.

*Back
to "Dynamic Geometry Sketches"*

*Back
to "Student Explorations"*

Created by Michael de Villiers, 14 August 2015.